Attractors and Expansion for Brownian Flows
Michael Scheutzow (Technische Universität Berlin)
Abstract
We show that a stochastic flow which is generated by a stochastic differential equation on $\mathbb{R}^d$ with bounded volatility has a random attractor provided that the drift component in the direction towards the origin is larger than a certain strictly positive constant $\beta$ outside a large ball. Using a similar approach, we provide a lower bound for the linear growth rate of the inner radius of the image of a large ball under a stochastic flow in case the drift component in the direction away from the origin is larger than a certain strictly positive constant $\beta$ outside a large ball. To prove the main result we use chaining techniques in order to control the growth of the diameter of subsets of the state space under the flow.
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Pages: 1193-1213
Publication Date: July 3, 2011
DOI: 10.1214/EJP.v16-894
References
- Arnold, Ludwig; Scheutzow, Michael. Perfect cocycles through stochastic differential equations. Probab. Theory Related Fields 101 (1995), no. 1, 65--88. MR1314175 (96b:60144)
- Borodin, Andrei N.; Salminen, Paavo. Handbook of Brownian motion—facts and formulae. Probability and its Applications. Birkhäuser Verlag, Basel, 1996. xiv+462 pp. ISBN: 3-7643-5463-1 MR1477407 (98i:60077)
- Chueshov, Igor; Scheutzow, Michael. On the structure of attractors and invariant measures for a class of monotone random systems. Dyn. Syst. 19 (2004), no. 2, 127--144. MR2060422 (2005c:37095)
- Cranston, Mike; Scheutzow, Michael. Dispersion rates under finite mode Kolmogorov flows. Ann. Appl. Probab. 12 (2002), no. 2, 511--532. MR1910637 (2003f:60075)
- Cranston, Mike; Scheutzow, Michael; Steinsaltz, David. Linear expansion of isotropic Brownian flows. Electron. Comm. Probab. 4 (1999), 91--101 (electronic). MR1741738 (2001d:60068)
- Cranston, Mike; Scheutzow, Michael; Steinsaltz, David. Linear bounds for stochastic dispersion. Ann. Probab. 28 (2000), no. 4, 1852--1869. MR1813845 (2001k:60087)
- Crauel, Hans; Dimitroff, Georgi; Scheutzow, Michael. Criteria for strong and weak random attractors. J. Dynam. Differential Equations 21 (2009), no. 2, 233--247. MR2506662 (2010h:37120)
- Crauel, Hans; Flandoli, Franco. Attractors for random dynamical systems. Probab. Theory Related Fields 100 (1994), no. 3, 365--393. MR1305587 (95k:58092)
- Flandoli, Franco; Schmalfuß, Björn. Weak solutions and attractors for three-dimensional Navier-Stokes equations with nonregular force. J. Dynam. Differential Equations 11 (1999), no. 2, 355--398. MR1695250 (2000j:60076)
- Kunita, Hiroshi. Stochastic flows and stochastic differential equations. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1990. xiv+346 pp. ISBN: 0-521-35050-6 MR1070361 (91m:60107)
- Lisei, Hannelore; Scheutzow, Michael. Linear bounds and Gaussian tails in a stochastic dispersion model. Stoch. Dyn. 1 (2001), no. 3, 389--403. MR1859014 (2002j:60102)
- Lisei, Hannelore; Scheutzow, Michael. On the dispersion of sets under the action of an isotropic Brownian flow. Probabilistic methods in fluids, 224--238, World Sci. Publ., River Edge, NJ, 2003. MR2083375 (2005f:37113)
- Ochs, Gunter. Weak random attractors. Institut für Dynamische Systeme, Universität Bremen, Report 449, 1999.
- Scheutzow, Michael. Comparison of various concepts of a random attractor: a case study. Arch. Math. (Basel) 78 (2002), no. 3, 233--240. MR1888707 (2002m:37072)
- Scheutzow, Michael. Attractors for ergodic and monotone random dynamical systems. Seminar on Stochastic Analysis, Random Fields and Applications V, 331--344, Progr. Probab., 59, Birkhäuser, Basel, 2008. MR2401964 (2009i:37129)
- Scheutzow, Michael. Chaining techniques and their application to stochastic flows. Trends in stochastic analysis, 35--63, London Math. Soc. Lecture Note Ser., 353, Cambridge Univ. Press, Cambridge, 2009. MR2562150 (2011c:60210)
- Scheutzow, Michael; Steinsaltz, David. Chasing balls through martingale fields. Ann. Probab. 30 (2002), no. 4, 2046--2080. MR1944015 (2003k:60166)

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